Very Large Numbers
For much of human history people needed only a few numbers. Some societies had words for “one” and for “many” and didn’t bother naming other values. Gradually, markers for larger quantities have evolved. Fingers and toes sufficed for a long time. Pebbles held in the hand, notches on a stick, or knots tied in a rope, could keep track of things such as months, cattle, baskets and neighbors. Then symbols were contrived and written down. Now, in the first part of the twenty-first century, large numbers are everywhere.
We hear about sums of money in the billions and trillions. We know these are big numbers and that trillions are bigger than billions, but most of us have difficulty understanding these enormous values. We have no trouble mentally comparing a ten dollar purchase and a ten-thousand dollar one. But what is a trillion dollar project? Or a multi-trillion dollar merger? Even the population of the United States is problematic. Three hundred thirty million, how much is that? With big numbers, we are similar to those primitive ancestors with their “one” and “many” differentiation.
Here is a way of thinking about large numbers that provides an intuitive comparison between the enormous magnitudes popular in our modern times.
This idea doesn't originate here. It is believed to have come from the book, “The Anthropic Cosmological Principle,” by John D. Barrow and Frank J. Tipler. Unfortunately, the exact citation cannot now be found without a rereading of the book.
Before we start, be advised we will quickly need to start using scientific notation. That’s a fancy term for counting zeros. 103 means a 1 followed by 3 zeros, also called one thousand. 106 means a 1 followed by 6 zeros, also called one million, and so on and so forth. Now, let’s start.
Consider the meter stick. It has one thousand, 103, millimeters ticked off along its length. Imagine a tiny ball bearing at each millimeter tick mark. We have one thousand, 103, ball bearings. So, in our conceptual construct, we will think of one meter stick as one thousand.
|Now, imagine a square, one meter on each side, with one thousand rows of those tiny ball bearings. This square will contain one million ball bearings, 106. Our model for one million, is thus, a square, one meter on a side. So far, so good. It’s pretty easy to imagine a square that is about one yard on each side. It fits easily into the living room.|
Now, lets stack one thousand of those squares of ball bearings, one on top of the other. We now have a cube, one meter on each side. It still fits nicely into the living room and it contains one billion, 109,ball bearings. Good, we’re up to a billion already, and still sitting in the living room.
Now, to paraphrase Everet Dirkson, the late senate minority leader, a billion here, a billion there, and we’re getting into some big numbers. Imagine a row of those one-meter cubes, one thousand cubes in all. This time we will have to move out of the living room. The row will be one thousand meters long, and will contain one trillion ball bearings, 1012. One thousand meters is also called a kilometer, and extends out of the living room, about half a mile down the street. That’s the difference between one billion and one trillion. One is in your living room and the other requires a hike across the neighborhood.
If we consider a square of our one trillion ball bearing rows. We’re now taking up several city blocks and we’ve gotten to 1015, one thousand trillion, also called a quadrillion, here in America. By the way, scientists estimate the number of cells in the human body, as well as the neuronal connections in the human brain, is about 1014. Think of it as 100 rows of the one-meter cubes.
Complete the cube made up of one thousand of those quadrillion ball bearing squares and we have a cube a little more than half a mile on a side, and we are now at 1018, also called a quintillion.
Extend a row of one thousand of our quintillion ball bearing cubes about six hundred miles down the road and we get to 1021, one sextillion. It would extend almost across the state of Texas.
One septillion, 1024, is a six hundred mile square of the sextillion ball bearing rows, almost covering the state of Texas.
One octillion, 1027, is a six hundred mile cube of our one septillion squares, allowing Houston to build a stairway three times higher than the International Space Station.
is a six hundred thousand mile row of our octillion
squares and extends off the planet to a point about 381,000 miles beyond the
One decillion, 1033, is a square of those six hundred thousand mile rows, extending around the earth, and out into the asteroid belt.
One undecillion, 1036, is a six hundred thousand mile cube engulfing the earth and surrounding space.
is a six hundred million mile row of
undecillions, extending past Jupiter, and to within 200 million miles of
One tredecillion, 1042, completes the square that covers the orbit of Jupiter but still stays inside Saturn.
One quattuordecillion, 1045, completes the cube inside Jupiter’s orbit. Ten of these squares, 1043, represents the so-called “Shannon number,” the lower bound on the game-tree complexity of chess.
One quindecillion, 1048, is a row of quattuordecillions, extending about 600 billion miles into space, or about one tenth of a light year. That is about 171 times the distance from Earth to Pluto.
One sexdecillion, 1051, completes the 600 billion mile square.
One septendecillion, 1054, completes the 600 billion mile cube.
One octodecillion, 1057, is a row of septendecillions extending about one hundred light years from earth. That’s about a tenth of the distance across our galaxy, the Milky Way.
One novemdecillion, 1060, is a square of octodecillions.
One vigintillion, 1063, is a cube of novemdecillions encompassing about 512 of what is known as illumination-type G stars.
One unvigintillion, 1066, is a row of vigintillions, extending about one hundred thousand light years from earth, which takes us across the Milky Way.
One duovigintillion, 1069, is a square of unvigintillions covering a one hundred thousand light year area, nearly all of the Milky Way.
One trevigintillion, 1072, is a cube of duovigintillions filling the space inside the Milky Way.
One quattuorvigintillion, 1075, is a row of trevigintillion cubes, extending about 100 million light years from earth, which takes us out beyond the Virgo Cluster of galaxies and three fourths of the way to the Great Attractor, a gravity anomaly in deep space discovered by the Hubble Space Telescope.
One quinvigintillion, 1078, is a square of quattuorvigintillions, covering 100 million light years.
One sexvigintillion, 1081, is a cube of quinvigintillions, filling a space 100 million light years on a side. This is also the estimated number of atoms in the observable universe.
One septenvigintillion, 1084, is a row of sexvigintillion cubes extending 100 billion light years, taking us beyond the observed universe. That’s far enough for now.
In trying to make sense of large numbers we’ve entered a conceptual area of difficulty with great distances. Still, the method does give us a relative sense of how much bigger each new multiple of one thousand becomes.For some further facts about correspondences between these large numbers and physical enumerations, see this post on the blog, Wait But Why. You will find out about some truly big numbers, numbers so big they intimidate us about using the word "infinite." How can we talk about "infinite" when we can't comprehend the finite?
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